# Science:Math Exam Resources/Courses/MATH101/April 2015/Question 05 (b)/Solution 1

Since is positive for any , monotonically decreasing, and converges to 0, applying the Alternating Series Estimation Theorem, we have

Now, it is enough to find the smallest such that

By taking square roots on both side, the inequality is equivalent with and hence with . Therefore, we can easily obtain the smallest natural number as 4 satisfying .

The answer: and the partial sum is